Effect of punching shear on load–deformation behavior of flat slabs
Introduction
Reinforced concrete (RC) flat slabs supported by columns are widely used in structural engineering as they are easy to build, therefore cost-efficient, and provide greater architectonical flexibility compared to joist constructions. Effective slab design has to ensure flexural capacity by providing adequate longitudinal reinforcement and sufficient punching resistance to bear the local concentration of shear loads in the column vicinity, e.g. via the placing of shear reinforcement. Additionally, the serviceability has to be ensured by e.g. limiting deformations or crack widths. In many cases the punching resistance is the decisive design criterion, in particular for elevated flexural reinforcement ratios. However, mutual interaction between both failure mechanisms has been observed. The punching shear resistance is therefore described as a function of the flexural resistance [1], [2], [3], [4] or as a moment–shear interaction relationship [5], [6], [7], [8].
The sector model, first introduced by Kinnunen and Nylander [9] and later adopted e.g. by Shehata [10], [11], [12], Gomes [13], [14], [15], or Muttoni [16] (when deriving his Critical Shear Crack Theory, CSCT), applies the plasticity theory to rotation-symmetric rigid slab segments rotating around the edge of a central column, see Fig. 1a. The model assumes rotation-symmetric reinforcement and establishes moment equilibrium on the slab segment. In the case of orthogonal reinforcement, the flexural stiffness is reduced since the directions of the principal moments may deviate from the rebar directions and a reduction of the torsional to the flexural stiffness ratio of the slab occurs due to concrete cracking [16], [17]. This stiffness reduction is taken into account by limiting the reinforcing steel’s Young’s modulus, Es, by a reduction factor, βE, for which numerous values are suggested [16], [17], [18], [19], [20].
Recently, Lips’ [17] experiments on square slabs with high shear reinforcement ratios showed a less stiff slab behavior and reduced flexural capacity compared to predictions by the CSCT. On the other hand, Guandalini’s [18] slabs, without shear reinforcement, did not exhibit any reduction of flexural capacity for cases with minimum longitudinal reinforcement. Slabs with higher longitudinal reinforcement ratios, however, failed in punching at loads and deformations much lower than at flexural failure. According to Lips, the reduced flexural capacity was a consequence of large shear deformations in the column vicinity. Since the influence of shear forces on deformation behavior is neglected in the original sector model, he proposed an iterative model for slabs with shear reinforcement considering a shear-critical region next to the column by adding a wedge (Fig. 1b) and limiting the radial curvatures inside the latter. The reducing effect of shear on the flexural capacity, however, had already been recognized before [21], [22]. Based on plasticity theory, a limitation of the bending moments was suggested to take the effect into account, but information on the deformation behavior was not provided since elastic deformations were neglected.
In the following, the rotation-symmetric Quadrilinear Sector Model [16] is firstly summarized and both effects are discussed: orthogonal reinforcement leading to reduced slab stiffness, and shear effect causing reduced flexural capacity. To take into account the first effect, a reduction factor of the flexural stiffness, βEI, is then derived (instead of a reduction factor of the reinforcement’s Young’s Modulus [16]) based on the Linear Compression Field Theory (LCFT) developed by Kupfer [23]. A Modified Sector Model (MSM) is subsequently proposed, in which a strength reduction factor, κV, for the longitudinal reinforcement, which crosses the shear crack, is introduced. This factor takes the shear effect into account and depends on the mechanical reinforcement ratio of the longitudinal reinforcement, ω. The MSM is validated using a set of experiments from literature, which was selected to cover an adequate variation of slab geometry, reinforcement ratio and material properties. The new model can also be applied to existing flat slabs that are strengthened against punching failure.
Section snippets
The sector model
In Kinnunen and Nylander’s [9] sector model, the rigid slab segments rotate around the column edges with a rotation angle, ψ. The segments are formed by one tangential shear crack and radial cracks. The assumption of a conical slab shape of the segment outside the shear crack geometrically defines the tangential curvature, thus χt = ψ/r (r = radius from slab center). Subsequently equilibrium conditions can be applied to the segment, depending on the constitutive relationship. Kinnunen and Nylander
Flexural stiffness reduction factor
Based on Villiger’s approach [20], a reduction factor for the flexural stiffness, βEI, is directly determined. With the assumption ρm = ρx = ρy and dm = dx = dy, the parameters ρm, dm, n, and from Eq. (5) can be inserted in Eqs. (A.4), (A.5), resulting in a flexural stiffness EIII = f(). By expressing the ratio of the flexural stiffness in principal direction 1 to that in x-direction, a directional flexural stiffness reduction factor, , results which is plotted in Fig. 6a. Unlike in Fig. 4b,
Strengthening concept
A strengthening method for existing flat slabs consisting of non-laminated and prestressed carbon fiber-reinforced polymer (CFRP) straps [52], [53] installed crosswise around the column was recently presented by the authors [54], [55], [56]. The four straps are installed in predrilled and precut openings and are anchored and prestressed from the bottom side of the slab. Two different anchor systems are available: the first system (slabs So1–4) consists of eight steel anchors adhesively bonded
Conclusions
The Quadrilinear Sector Model (QSM) allows the modeling of the load–rotation responses of reinforced concrete flat slabs. To take into account an orthogonal reinforcement layout and the associated increased slab deformations, the steel Young’s Modulus is normally reduced by a factor, βE. The QSM also neglects shear deformations, which may reduce the flexural capacity. The work presented here introduces the following improvements of the QSM:
- 1.
Instead of the mechanically not well-justified Young’s
Acknowledgments
The authors wish to acknowledge the support and funding of the Swiss Federal Commission for Technology and Innovation CTI (Grant No. 11569.1 PFIW-IW) and of F.J. Aschwanden AG, Lyss, Switzerland.
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